Interesting. A couple of times now, a hummingbird has had its beak stuck through my screen and I've had to (very gently) push it out. I never thought about going to the outside of the screen and pulling it out, thinking I would put it into even more of a panic.
I'd probably do the same thing in your situation. I grew up with birds (parakeets, cockatiels, parrots), so I didn't even think twice about handling it safely. In fact one parakeet we got for free from K Mart back when they sold birds. It was loose in the store. My father offered to catch it, and the manager said, "if you catch it you can keep it." And thus
Princess became a member of our little zoo.
It may not be safe to say this on a board of skeptics, but I firmly believe that some people just have a way with animals. I seem to be one of them. For whatever reason I've always found myself capable of calming excited animals and approaching scared ones without terrifying them. It's one of those things where it wasn't until I was older and more experienced that I realized this was not typical.
I remember a few years ago a friend had young dog (teenager equivalent) that was rather wild. We were at his house, and I was playing a little game of chase with the dog. He was running around in circles and approaching just close enough so I could playfully smack him on his rump sometimes. He loved it. My friend said, "Oh, great. Now he's never gonna calm down."
I said, "Sure he will." I stopped the game, and the dog did a few more laps trying to get it going again. I calmly told him we were done and that he needed to come sit with me. Sure enough, he calmed down and came over. I told him to sit (once), and he sat and leaned up against my leg.
My friend was astonished. "He hardly ever sits when we tell him to. And once he's riled up like that he's all over the place." He wanted to know how I did it, but I can't really explain it. It seems to be something in my demeanor and my voice.
It was kind of the same thing with the hummingbird. It was frantic trying to fly into the kitchen, sort of like a moth trying to get into the house, repeatedly banging into the window. In fact, when I first heard it I though it was one of those
giant moths I regularly see here in Phoenix. I just talked to it and let it gently get into my hand. It didn't struggle at all and barely complained vocally as I cupped my other hand to keep it from taking off under the covered patio. Then again, maybe hummingbirds are cool with humans. I got one to land on me a few years ago.